Right triangle calculator (c,v)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered hypotenuse c and height h.

Right scalene triangle.

The lengths of the sides of the triangle:
a = 37.88334384895
b = 9.2655262447
c = 39

Area: T = 175.5
Perimeter: p = 86.14987009365
Semiperimeter: s = 43.07443504683

Angle ∠ A = α = 76.25767868748° = 76°15'24″ = 1.3310932008 rad
Angle ∠ B = β = 13.74332131252° = 13°44'36″ = 0.24398643188 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Altitude (height) to the side a: ha = 9.2655262447
Altitude (height) to the side b: hb = 37.88334384895
Altitude (height) to the side c: hc = 9

Median: ma = 21.0866341934
Median: mb = 38.16656414048
Median: mc = 19.5

Line segment ca = 2.2011156108
Line segment cb = 36.7998843892

Inradius: r = 4.07443504683
Circumradius: R = 19.5

Vertex coordinates: A[39; 0] B[0; 0] C[36.7998843892; 9]
Centroid: CG[25.26662812973; 3]
Coordinates of the circumscribed circle: U[19.5; -0]
Coordinates of the inscribed circle: I[33.80990880213; 4.07443504683]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 103.74332131252° = 103°44'36″ = 1.3310932008 rad
∠ B' = β' = 166.25767868748° = 166°15'24″ = 0.24398643188 rad
∠ C' = γ' = 90° = 1.57107963268 rad


Calculate another triangle

How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other triangle characteristics, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Input data entered: hypotenuse c and height h

c=39 h=9c = 39 \ \\ h = 9

2. From the hypotenuse c and height h, we calculate a,b - Pythagorean theorem, Euclid's theorem:

 c=c1+c2 h2=c1⋅ c2  h2=c1⋅ (c−c1) h2=c1⋅ c−c12  c12−c1⋅ c+h2=0  c12−39⋅ c1+81=0  c1=36.799 c2=2.201  a=c12+h2=36.7992+92=37.883 b=c22+h2=2.2012+92=9.265 \ \\ c = c_1+c_2 \ \\ h^2 = c_1 \cdot \ c_2 \ \\ \ \\ h^2 = c_1 \cdot \ (c-c_1) \ \\ h^2 = c_1 \cdot \ c-c_1 ^2 \ \\ \ \\ c_1^2 -c_1 \cdot \ c + h^2 = 0 \ \\ \ \\ c_1^2 -39 \cdot \ c_1 + 81 = 0 \ \\ \ \\ c_1 = 36.799 \ \\ c_2 = 2.201 \ \\ \ \\ a = \sqrt{ c_1^2+h^2 } = \sqrt{ 36.799^2+9^2 } = 37.883 \ \\ b = \sqrt{ c_2^2+h^2 } = \sqrt{ 2.201^2+9^2 } = 9.265

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides (SSS).
a=37.883 b=9.265 c=39a = 37.883 \ \\ b = 9.265 \ \\ c = 39

3. The triangle perimeter is the sum of the lengths of its three sides

p=a+b+c=37.883+9.265+39=86.149p = a+b+c = 37.883+9.265+39 = 86.149

4. The semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles to be given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s=p2=86.1492=43.074s = \dfrac{ p }{ 2 } = \dfrac{ 86.149 }{ 2 } = 43.074

5. The triangle area - from two legs

T=ab2=37.883⋅ 9.2652=175.5T = \dfrac{ ab }{ 2 } = \dfrac{ 37.883 \cdot \ 9.265 }{ 2 } = 175.5

6. Calculate the heights of the right triangle from its area.

ha=b=9.265  hb=a=37.883  T=chc2   hc=2 Tc=2⋅ 175.539=9h _a = b = 9.265 \ \\ \ \\ h _b = a = 37.883 \ \\ \ \\ T = \dfrac{ c h _c }{ 2 } \ \\ \ \\ \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 175.5 }{ 39 } = 9

7. Calculation of the inner angles of the triangle - basic use of sine function

sin⁡α=ac α=arcsin⁡(ac)=arcsin⁡(37.88339)=76°15′24" sin⁡β=bc β=arcsin⁡(bc)=arcsin⁡(9.26539)=13°44′36" γ=90°\sin α = \dfrac{ a }{ c } \ \\ α = \arcsin(\dfrac{ a }{ c } ) = \arcsin(\dfrac{ 37.883 }{ 39 } ) = 76\degree 15'24" \ \\ \sin β = \dfrac{ b }{ c } \ \\ β = \arcsin(\dfrac{ b }{ c } ) = \arcsin(\dfrac{ 9.265 }{ 39 } ) = 13\degree 44'36" \ \\ γ = 90\degree

8. Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of a triangle's inradius and semiperimeter (half the perimeter) is its area.

T=rs r=Ts=175.543.074=4.074T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 175.5 }{ 43.074 } = 4.074

9. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R=2c​=239​=19.5

10. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio of 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate a median's length from its side's lengths.

ma2=b2+(a/2)2 ma=b2+(a/2)2=9.2652+(37.883/2)2=21.086  mb2=a2+(b/2)2 mb=a2+(b/2)2=37.8832+(9.265/2)2=38.166  mc=R=c2=392=19.5m_a^2 = b^2 + (a/2)^2 \ \\ m_a = \sqrt{ b^2 + (a/2)^2 }= \sqrt{ 9.265^2 + (37.883/2)^2 } = 21.086 \ \\ \ \\ m_b^2 = a^2 + (b/2)^2 \ \\ m_b = \sqrt{ a^2 + (b/2)^2 }= \sqrt{ 37.883^2 + (9.265/2)^2 } = 38.166 \ \\ \ \\ m_c = R = \dfrac{ c }{ 2 } = \dfrac{ 39 }{ 2 } = 19.5

Calculate another triangle

Look also at our friend's collection of math problems and questions:

See triangle basics on Wikipedia or more details on solving triangles.